Calculus AB: Applications of the Derivative

Problems for "Using the Second Derivative to Analyze Functions"

Vertical and Horizontal Asymptotes, page 2

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Vertical Asymptotes

A vertical asymptote occurs at
x = c
when the following are all true

1) f (c) is undefined

2) f (x) = ∞ or - ∞

3) f (x) = ∞ or - ∞

Taken together, #2 and #3 mean that
f
"grows without bound" as it approaches
x = c
. This happens most often with a rational function at a value of
x
that
leads to a denominator of zero. For example, consider
f (x) =
.
f (x)
is undefined
at
x = - 1
.

1) f (x) is undefined at x = - 1

2) = - ∞

3) = + ∞

Thus,
x = - 1
is a vertical asymptote of
f
, graphed below:

Figure %:
f (x) =
has a vertical asymptote at
x = - 1

Horizontal Asymptotes

A horizontal asymptote is a horizontal line that the graph of a function approaches,
but never touches as
x
approaches negative or positive infinity.
If
f (x) = L
or
f (x) = L
, then the line
y = L
is a horiztonal asymptote of the function
f.
For example, consider the function
f (x) =
.
This function has a horizontal asymptote at
y = 2
on both the left and the right ends of
the graph:

Figure %:
f (x) =
. Has a horizontal asymptote at
y = 2

Note that a function may cross its horizontal asymptote near the origin, but it cannot cross
it as
x
approaches infinity.

Intuitively, we can see that
y = 2
is a horizontal asymptote of
f
because as
x
approaches infinity,
f (x) =
behaves more and more like
f (x) =
, which is the same as
f (x) = 2
. Although
f
behaves more and
more like this, it never actually becomes this function, so
y = 2
is approached but not
reached.